Surely it is different from a sense of self. Your sense of self is inherently linked to the memories and experiences you have ‘stored’ in this body, the neuron pathways you have hard wired through use and practice and certain motor skills you accept autonomously.
My wife made just this point – but I don’t think you are any less “you” at birth when you have almost zero memories/experiences.
The sense of “self”, to me, is about the fact that of all the consciousnesses there are strolling around on the planet, there’s only one which you see from the inside as it were. Which, as user-removed points out, some would call the soul. If you reject the idea of the soul, it seems to be that if you accept you might ever see another consciousness from the inside – a kind of non-spiritual reincarnation – you would have to accept that it could happen twice at the same time. That’s really mind-boggling and impossible. Therefore, by the power of logic, I have to accept that this is the only consciousness I will ever inhabit (or accept the idea of a soul floating from one body to another which I also reject).